Question

The eigenstates and eigenenergies for a system of Bloch electrons satisfy the following H(k)|Unk) = En(k)|Unk); Ink) = eik:x|unk), where n is a band index and k is the wave vector spanning the first Brillouin zone. The Bloch states (Unk) are periodic reflecting the periodicity of the system for the electrons. How does such Hamiltonian transform under time reversal operation? Obtain the condition under which such a Hamiltonian is invariant under time reversal operation.

Answer 1

H(K)14hk) = E(k) lok/ where I Unk>= pikx unk? HCK) eikxUnx > = Enck) e' kx Unk) Ô → Time Reversal operator. @lynk) - ) [~ represents tilta i-e time reversed State. Tune - Tegne o que eu se Ipink) -86 = (1 - inst) | ynx)=0 Ô [(1 -4 (-86] [ 4 x 2 = 0 = 1 rank 7-81 (1 - it st)&nk? -0 = lymieeot (1-# ) 61 yax) -o = @[149 (30) Myto so - THÔ= ô (H) - 0 or, - Ĥ ôl Ink >= 6 i lunk? 08 - û {ol und >} = En {olunk >} oo, û şôl Unk>} = - En{ê lunk}} This does not gives any sense.

f | Unk>= En | Unk> We have one theorem relating to Antilinear opere À (619)) = (*216) Thus in this egn @ we can't take out then it would be ih ô = ô Cih) or i hô=-io or ho = ô [ Å Ô ] = 0 This Hamiltonian commute under time reversal operation. Now for Non-degenerate eigen stale. û In>= Eniny We have h ô = ô So, û şê In)} = En {ô In>} In>= oln) Talny = CalIn>= <alny so, Y. (+) = 4 * (x) Unk (x) = Unkn) This implies that if Hamiltonian is invariant and Our system is non degenerate then our eigen Functions are always real. This is true for spinies ô ln>= eisin) 0²/n) = Ô {eis In>} = 10) Thus In> and Olny are degenerate Here any is nothing but i e unk) and S= kx. sparticles.